Lattice Coverings and Gaussian Measures of n-Dimensional Convex Bodies

Let ‖ · ‖ be the euclidean norm on R and γn the (standard) Gaussian measure on R with density (2π)e 2/2. Let θ (≃ 1.3489795) be defined by γ1([−θ/2, θ/2]) = 1/2 and let L be a lattice in R n generated by vectors of norm ≤ θ. Then, for any closed convex set V in R with γn(V ) ≥ 1 2 and for any a ∈ R, (a + L) ∩ V 6= φ. The above statement can be viewed as a “nonsymmetric” version of Minkowski Theorem. Let U , V be a pair of convex sets in R containing the origin in the interior. Let us define β(U, V ) as the smallest r > 0 satisfying the following condition: to each sequence u1, . . . , un ∈ U there correspond signs ε1, . . . , εn = ±1 such that ε1u1 + · · ·+ εnun ∈ rV . Upper and lower bounds for β(U, V ) for various sets U and V (usually centrally symmetric) were investigated by several authors. We will mention some of their results once the appropriate notation is introduced, see also references in [3]. Let L be a lattice in R, i.e. an additive subgroup of R generated by n linearly independent vectors. The quantities (again, usually defined for centrally symmetric sets) λn(L,U) = min{r > 0 : dim span (L ∩ rU) = n}, μ(L, V ) = min{r > 0 : L+ rV = R} are called the nth minimum and the covering radius of L with respect to U and V , respectively; sometimes μ(L, V ) is called ”the nth covering minimum” and denoted μn(L, V ). Let us define AMS Subject Classification 11H06, 11H31, 52C07, 52C17 ∗Part of this research was done while this author was visiting Case Western Reserve University under a cooperation grant from KBN (Poland) and NSF (U.S.A.) †Supported in part by the National Science Foundation.